The Plasticity of Some Fittable Surfaces on a Given Quadruple of Points in the Three-Dimensional Euclidean Space
We construct a two-dimensional sphere in the three-dimensional Euclidean space which intersects a circular cylinder in three given points and the corresponding weighted Fermat-Torricelli point for a geodesic triangle such that these three points and the corresponding weighted Fermat- Torricelli poin...
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irk-123456789-1068102016-10-06T03:02:30Z The Plasticity of Some Fittable Surfaces on a Given Quadruple of Points in the Three-Dimensional Euclidean Space Zachos, A.N. We construct a two-dimensional sphere in the three-dimensional Euclidean space which intersects a circular cylinder in three given points and the corresponding weighted Fermat-Torricelli point for a geodesic triangle such that these three points and the corresponding weighted Fermat- Torricelli point remain the same on the sphere for a different triad of weights which correspond to the vertices on the surface of the sphere. We derive a circular cone which passes from the same points that a circular cylinder passes. By applying the inverse weighted Fermat-Torricelli problem for different weights, we obtain the plasticity equations which provide the new weights of the weighted Fermat-Torricelli point for fixed geodesic triangles on the surface of a fittable sphere and a fittable circular cone with respect to the given quadruple of points on a circular cylinder, which inherits the curvature of the corresponding fittable surfaces. Построена двумерная сфера в трехмерном евклидовом пространстве, которое пересекает круговой цилиндр в трех заданных точках и соответствующей взвешенной точке Ферма-Торричелли для геодезического треугольника так, что эти три точки и соответствующая взвешенная точка Ферма-Торричелли остаются такими же на сфере и для другой триады весов, которые соответствуют вершинам на поверхности сферы. Выведен круговой конус, который проходит через те же точки, что и круговой цилиндр. Применяя обратную взвешенную Ферма-Торричелли задачу для различных весов, получаем уравнения пластичности, которые обеспечивают новые веса для взвешенной точки Ферма-Торричелли для фиксированных геодезических треугольников на поверхности подходящей сферы и подходящего кругового конуса по отношению к данным четырем точкам на круговом цилиндре, который унаследует кривизну соответствующих подходящих поверхностей. 2014 Article The Plasticity of Some Fittable Surfaces on a Given Quadruple of Points in the Three-Dimensional Euclidean Space / A.N. Zachos // Журнал математической физики, анализа, геометрии. — 2014. — Т. 10, № 4. — С. 485-495. — Бібліогр.: 13 назв. — англ. 1812-9471 DOI: http://dx.doi.org/10.15407/mag10.04.485 http://dspace.nbuv.gov.ua/handle/123456789/106810 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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We construct a two-dimensional sphere in the three-dimensional Euclidean space which intersects a circular cylinder in three given points and the corresponding weighted Fermat-Torricelli point for a geodesic triangle such that these three points and the corresponding weighted Fermat- Torricelli point remain the same on the sphere for a different triad of weights which correspond to the vertices on the surface of the sphere. We derive a circular cone which passes from the same points that a circular cylinder passes. By applying the inverse weighted Fermat-Torricelli problem for different weights, we obtain the plasticity equations which provide the new weights of the weighted Fermat-Torricelli point for fixed geodesic triangles on the surface of a fittable sphere and a fittable circular cone with respect to the given quadruple of points on a circular cylinder, which inherits the curvature of the corresponding fittable surfaces. |
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Zachos, A.N. |
| spellingShingle |
Zachos, A.N. The Plasticity of Some Fittable Surfaces on a Given Quadruple of Points in the Three-Dimensional Euclidean Space Журнал математической физики, анализа, геометрии |
| author_facet |
Zachos, A.N. |
| author_sort |
Zachos, A.N. |
| title |
The Plasticity of Some Fittable Surfaces on a Given Quadruple of Points in the Three-Dimensional Euclidean Space |
| title_short |
The Plasticity of Some Fittable Surfaces on a Given Quadruple of Points in the Three-Dimensional Euclidean Space |
| title_full |
The Plasticity of Some Fittable Surfaces on a Given Quadruple of Points in the Three-Dimensional Euclidean Space |
| title_fullStr |
The Plasticity of Some Fittable Surfaces on a Given Quadruple of Points in the Three-Dimensional Euclidean Space |
| title_full_unstemmed |
The Plasticity of Some Fittable Surfaces on a Given Quadruple of Points in the Three-Dimensional Euclidean Space |
| title_sort |
plasticity of some fittable surfaces on a given quadruple of points in the three-dimensional euclidean space |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2014 |
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http://dspace.nbuv.gov.ua/handle/123456789/106810 |
| citation_txt |
The Plasticity of Some Fittable Surfaces on a Given Quadruple of Points in the Three-Dimensional Euclidean Space / A.N. Zachos // Журнал математической физики, анализа, геометрии. — 2014. — Т. 10, № 4. — С. 485-495. — Бібліогр.: 13 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT zachosan theplasticityofsomefittablesurfacesonagivenquadrupleofpointsinthethreedimensionaleuclideanspace AT zachosan plasticityofsomefittablesurfacesonagivenquadrupleofpointsinthethreedimensionaleuclideanspace |
| first_indexed |
2025-07-07T19:04:08Z |
| last_indexed |
2025-07-07T19:04:08Z |
| _version_ |
1837016078080802816 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2014, vol. 10, No. 4, pp. 485–495
The Plasticity of Some Fittable Surfaces on a Given
Quadruple of Points in the Three-Dimensional
Euclidean Space
A.N. Zachos
University of Patras, Department of Mathematics
GR-26500 Rion, Greece
E-mail: azachos@gmail.com
Received September 1, 2013, revised April 4, 2014
We construct a two-dimensional sphere in the three-dimensional Eu-
clidean space which intersects a circular cylinder in three given points and
the corresponding weighted Fermat–Torricelli point for a geodesic trian-
gle such that these three points and the corresponding weighted Fermat–
Torricelli point remain the same on the sphere for a different triad of weights
which correspond to the vertices on the surface of the sphere. We derive a
circular cone which passes from the same points that a circular cylinder
passes. By applying the inverse weighted Fermat–Torricelli problem for dif-
ferent weights, we obtain the plasticity equations which provide the new
weights of the weighted Fermat–Torricelli point for fixed geodesic triangles
on the surface of a fittable sphere and a fittable circular cone with respect
to the given quadruple of points on a circular cylinder, which inherits the
curvature of the corresponding fittable surfaces.
Key words: weighted Fermat–Torricelli point, sphere, circular cylinder,
circular cone, fittable surfaces.
Mathematics Subject Classification 2010: 51E12, 52A10, 52A55, 51E10.
1. Introduction
The weighted Fermat–Torricelli problem states that:
Given are three points A1, A2, A3 in the Euclidean plane, three positive real
numbers wi (weight) which correspond to the vertex Ai, find a point X in the
Euclidean plane that minimizes the sum of the weighted Euclidean distances
f(X) = w1‖A1X‖+ w2‖A2X‖+ w3‖A3X‖.
The solution of the weighted Fermat–Torricelli problem is named as the
weighted Fermat–Torricelli point F. E. Torricelli was the first to discover the
c© A.N. Zachos, 2014
A.N. Zachos
isogonal property (or 120◦ property) of the weighted Fermat–Torricelli point
∠A1FA2 = ∠A2FA3 = ∠A3FA1 = 120◦ for equal weights. B. Cavalieri was
the first who stated that if at most one angle ∠AiAjAk ≥ 120◦, then F = Aj for
w1 = w2 = w3, i, j, k = 1, 2, 3, i 6= j 6= k (see [2, 4]).
The isogonal property of the equally weighted Fermat–Torricelli point holds
in Riemmanian manifolds ([3]) and in an Alexandrov surface of the bounded
curvature ([5], in the surface of polyhedra).
We introduce a problem of the (curvature) plasticity of a surface which passes
from four given points in R3 :
Problem 1 (Problem of plasticity of fittable surfaces in R3). Suppose that
F is the corresponding weighted Fermat–Torricelli point of a geodesic triangle
4A1A2A3 on a C2 complete surface M with weights w1, w2, and w3. Find a
fittable Alexandrov surface M ′ of the bounded curvature which passes from A1,
A2, A3, and F such that F is the corresponding weighted Fermat–Torricelli point
of 4A1A2A3 on M ′ with weights w′1, w′2, and w′3.
In this paper, we apply the weighted Fermat-Torricelli problem for geodesic
triangles on certain surfaces in the three- dimensional Euclidean space, the inverse
weighted Fermat-Torricelli problem, in order to derive the equations which allow
us to compute the weights corresponding to the fittable surfaces for three fixed
points and a fixed fourth point (weighted Fermat-Torricelli point) located at
the interior of the geodesic triangle for the case of a two-dimensional sphere
in the three-dimensional Euclidean space which intersects a circular cylinder in
three given points and the corresponding weighted Fermat–Torricelli point for a
geodesic triangle and a fittable circular cone which passes from the same points
that a circular cylinder passes.
2. Plasticity of a Sphere and Circular Cone with Respect to a
Circular Cylinder in the Three-Dimensional Euclidean Space
Let 4 (A1A2A3)C be a geodesic triangle, for instance, on a circular cylinder
x2 + y2 = 1 for z1 ≤ z ≤ z2 and FC ≡ A0 ia the corresponding weighted Fermat–
Torricelli point for given weights w1, w2, and w3.
By Ai = (cosϕi, sinϕi, zi), we denote the points located on the circular
cylinder x2 + y2 = 1, by (aij)C , the length of the geodesic arc AiAj , by ~rij =
(cos t, sin t, bijt), the circular helix from Ai to Aj , by (αijk)C , the angle formed by
AiAj and AjAk, by Aip, the projection of Ai to the circle of the cylinder which
passes from A1 = (1, 0, 0), by ω0, the angle ∠A0pA1A0, by z0, the linear segment
A0A0p, and by L0, the linear segment A1A0p for i, j, k = 0, 1, 2, 3, i 6= j and
j 6= k.
We set b12 ≡ z2
ϕ2
and b13 =≡ z3
ϕ3
, where 0 < ϕi < π, for i, j = 1, 2, 3 and i 6= j.
486 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4
The Plasticity of Some Fittable Surfaces
We need the following two lemmata proved in [12] and [11] (see also in [12]):
Lemma 1. [12, Theorem 1, p. 173]. The exact location of the weighted
Fermat–Torricelli point A0 = A0(x0, y0, z0) of 4 (A1A2A3)C , composed of three
circular helixes on the circular cylinder, is given by the following three equations:
ω0 = arctan b12 + arccos
(
1 + b12b13√
1 + b2
12
√
1 + b2
13
)
− arccot
[(√√√√1−
(
1 + b12b13√
1 + b2
12
√
1 + b2
13
)2
− 1 + b12b13√
1 + b2
12
√
1 + b2
13
cot
(
arccos
w2
3 − w2
1 − w2
2
2w1w2
)
−
√
1 + b2
13ϕ3√
1 + b2
12ϕ2
cot
(
arccos
w2
2 − w2
1 − w2
3
2w1w3
))/
(
− 1 + b12b13√
1 + b2
12
√
1 + b2
13
−
√
1−
(
1 + b12b13√
1 + b2
12
√
1 + b2
13
)2
cot
(
arccos
w2
3 − w2
1 − w2
2
2w1w2
)
+
√
1 + b2
13ϕ3√
1 + b2
12ϕ2
)]
(2.1)
z0 =
sin
(
arctan b13 − ω0 + arccos w2
2−w2
1−w2
3
2w1w3
)√
1 + b2
13ϕ3
sin
(
arccos w2
2−w2
1−w2
3
2w1w3
) sinω0 (2.2)
and
L0 =
sin
(
arctan b13 − ω0 + arccos w2
2−w2
1−w2
3
2w1w3
)√
1 + b2
13ϕ3
sin
(
arccos w2
2−w2
1−w2
3
2w1w3
) cosω0. (2.3)
We consider the same points A1, A2, A3, and A0 on a sphere S(A0, R) and we
denote by 4 (A1A2A3)S the geodesic triangle on S(A0, R), by (aij)S , the length
of the geodesic arc AiAj , by (αijk)S , the angle formed by AiAj and AjAk and
w′i, the weight which corresponds to Ai and minimizes the objective function
w′1(a01)S + w′2(a02)S + w′3(a03)S for i, j, k = 0, 1, 2, 3 and i 6= j 6= k.
We set
ci ≡ sin(κ(ajk)S)
sin((αj0k)S)
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4 487
A.N. Zachos
for i, j, k = 1, 2, 3 and i 6= j 6= k, where
κ =
{ √
K if K = 1
R2 > 0,
i
√−K if K < 0.
Lemma 2. [13, Theorem 2.4, p. 115]. A finite set of solutions of the weighted
Fermat–Torricelli problem on the K-plane(two-dimensional sphere, hyperbolic
plane), which yields the global minimum point A0 (weighted Fermat–Torricelli
point), is given by the following equation with respect to the variable z = sin(α013)S :
c3
c2
± sin(α123)S
√
1−
(
c2z
c1
)2
− cos(α123)S
(
c2z
c1
)
= −c3
c1
sin(α213)S cos(α132)S
√
1− z2 +
c3
c1
cos(α213)S cos(α132)Sz
±(sinα132)S
√
1−
(
c3
c1
)2[
− sin(2(α213)S)z
√
1−z2 + cos(2(α213)S)z2 + sin2(α213)S
]
.
(2.4)
We recall the inverse weighted Fermat–Torricelli problem on a C2 surface in
R3 first stated by S. Gueron and R. Tessler in R2 ([2, 8, 9, 10]):
Problem 2. [2, p. 449], [8, Problem 3.2, p. 61] [9, Problem 2, p. 52], [10].
Given is a point A0 ∈ 4A1A2A3 on a C2 surface in R3. Does there exist a unique
set of positive weights wi, normalized by w1+w2+w3 = 1, for which A0 minimizes
w1(a01)g + w2(a02)g + w3(a03)g,
where (a0i)g is the length of the geodesic arc A0Ai?
Lemma 3. [2], [8, Proposition 3.2, Corollary 3.3, p. 61] [9, Proposition 5,
p. 52], [10]. The solution of the inverse weighted Fermat–Torricelli problem on a
C2 surface in R3 is given by
wi =
1
1 + sin αi0j
sin αj0k
+ sin αi0k
sin αj0k
(2.5)
for i, j, k = 1, 2, 3 and i 6= j 6= k.
We assume that Ai = (xi, yi, zi) and FC = FS ≡ A0 = (xF , yF , zF ) are located
at the intersection of the circular cylinder C and the sphere S(x0, y0, z0; R) for
i = 1, 2, 3.
488 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4
The Plasticity of Some Fittable Surfaces
Theorem 1. The following equations provide the plasticity of a sphere derived
by a circular cylinder with respect to the fixed points {A1A2A3A0} for a different
triad of weights w1, w2, w3, and w′1, w′2, w′3 such that FC = FS ≡ A0 :
wi =
1
1 + sin (αi0j)S
sin (αj0k)S
+ sin (αi0k)S
sin (αj0k)S
, (2.6)
for i, j, k = 1, 2, 3 and i 6= j 6= k , the angles (αi0j)S are determined by the
equations
x0 = −−d3g2h1 + d2g3h1 + d3g1h2 − d1g3, h2 − d2g1h3 + d1g2h3
f3g2h1 − f2g3h1 − f3, g1h2 + f1g3h2 + f2g1h3 − f1g2h3
, (2.7)
y0 = −d3f2h1 − d2f3h1 − d3f1h2 + d1f3h2 + d2f1h3 − d1f2h3
f3g2h1 − f2g3h1 − f3g1h2 + f1g3h2 + f2g1h3 − f1g2h3
, (2.8)
z0 =
d3f2g1 − d2f3g1 − d3f1g2 + d1f3g2 + d2f1g3 − d1f2g3
f3g2h1 − f2g3h1 − f3g1h2 + f1g3h2 + f2g1h3 − f1g2h3
, (2.9)
w′1 + w′2 + w′3 = 1, (2.10)
where
f1 = (xi − xF ), (2.11)
g1 = (yi − yF ), (2.12)
h1 = (zi − zF ), (2.13)
and
di = 0.5[(xi − xF )(xi + xF ) + (yi − yF )(yi + yF ) + (zi − zF )(zi + zF )]. (2.14)
P r o o f. Let 4 (A1A2A3)C be a geodesic triangle which is composed
of three circular helixes on a circular cylinder x2 + y2 = 1 for z1 ≤ z ≤ z2,
and FC be the corresponding weighted Fermat–Torricelli point. By unrolling the
cylinder, we get an isometric mapping of 4A1A2A3 to the Euclidean plane R2.
From Lemma 1, we derive the exact location of FC = (xF , yF , zF ).
We construct a sphere S(A0(x0, y0, z0), R) which passes from A1 = (x1, y1, z1),
A2 = (x2, y2, z2), A3 = (x3, y3, z3) and F = (xF , yF , zF ).
The bisectors of the linear segments AiF pass from Mi = (xi+xF
2 , yi+yF
2 , zi+zF
2 )
and intersect at A0 = (x0, y0, z0), such that ‖AiA0‖ = R, for i = 1, 2, 3 (Fig. 1).
Thus, we get
(xi − xF )x + (yi − yF )y + (zi − zF )z
= 0.5[(xi − xF )(xi + xF ) + (yi − yF )(yi + yF ) + (zi − zF )(zi + zF )] (2.15)
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4 489
A.N. Zachos
-5
0
5
x
-5 0
5y
-5
0
5
z
Fig. 1.
for i = 1, 2, 3. The intersection of the three planes (2.15) gives (2.7), (2.8), and
(2.9).
Thus, we get
R =
1√
K
=
√
(xi − x0)2 + (yi − y0)2 + (zi − z0)2, (2.16)
cos θiF = 1− 1
2
(
(xi − xF )2 + (yi − yF )2 + (zi − zF )2
R
)2
, (2.17)
(aiF )S = RθiF , (2.18)
and
(aij)S = Rθij (2.19)
for i, j = 1, 2, 3, i 6= j.
Therefore, the angles (αi0j)S are determined by the spherical cosine law in
4AiAjAk :
(αi0j)S = arccos
cosκ(aij)S − cosκ(a0i)S cosκ(a0j)S
sinκ(a0i)S sinκ(a0j)S
. (2.20)
Then, by applying Lemma 3, we obtain (2.6).
490 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4
The Plasticity of Some Fittable Surfaces
-2
-1
0
1
2
x
-2
-1
0
1
2
y -2
-1
0
1
2
z
Fig. 2.
R e m a r k 1. We note that there is a particular case where A1, A2, A3, and
F can be located on a circular cylinder x2 + y2 = Rx with the radius R
2 and on
a sphere x2 + y2 + z2 = R2 with the radius R. The intersection of this circular
cylinder and the sphere is called a Viviani curve with one point of self-intersection
(Fig.2, [6, Example 1.2.4 (a), p. 5]).
We consider the intersection of a circular cylinder C : x2 + y2 = 1 for z1 ≤
Z ≤ z2 and a circular cone Co : (x − x0)2 + (z − z0)2 =
(
r1
H
)2 (z −H)2. By H,
we denote the height of the circular cylinder and by r1, the radius of the circle
which corresponds to the basis of the circular cone.
Theorem 2. The following equations provide the plasticity of a circular cone
derived by a circular cylinder with respect to the fixed points {A1A2A3A0} for a
different triad of weights w1, w2, w3, and w′1, w′2, w′3 such that FCo = FC ≡ A0 :
wi =
1
1 + sin (αi0j)Co
sin (αj0k)Co
+ sin (αi0k)Co
sin (αj0k)Co
, (2.21)
and the angles (αi0j)S are determined by the equations
(x1 − x0)2 + (z1 − z0)2 =
(r1
H
)2
(z1 −H)2, (2.22)
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4 491
A.N. Zachos
(x2 − x0)2 + (z2 − z0)2 =
(r1
H
)2
(z2 −H)2, (2.23)
(x3 − x0)2 + (z3 − z0)2 =
(r1
H
)2
(z3 −H)2, (2.24)
(xF − x0)2 + (zF − z0)2 =
(r1
H
)2
(zF −H)2, (2.25)
where
w′1 + w′2 + w′3 = 1. (2.26)
P r o o f. By considering a fittable circular cone Co : (x− x0)2 + (z− z0)2 =(
r1
H
)2 (z − H)2, which passes from the points A1, A2, A3, and FCo ≡ FC = A0,
we get the system of equations (2.22), (2.23), (2.24), and (2.25) with respect to
the four variables x0, y0, r1, and z0 = H, which can give numerically the vertex
A of the circular cone.
Then, by unrolling the circular cone Co along A1A, we derive an isometric
mapping from Co to R2, which determines the angles (αijk)0 = (αijk)Co, and
obtain (2.21).
Taking into account that A is the vertex of the circular cone, r1 is the radius
of the circle c(P, r1) at the basis of the cone, H is the height of the cone, we
denote by ϕ0 the angle ∠A1PA0p, where A0p is the point of intersection of AA0
and c(P, r1), and by x00, the length of the linear segment A0A, and we consider
the lemma proved in ([12]).
Lemma 4. [12, Theorem 2, p. 177–178]. The exact location of the weighted
Fermat–Torricelli point FCo of 4A1A2A3 on Co is given by the following two
equations:
x00 =
√
(1 + H2) + (a01)2g − 2
√
1 + H2(a01)g cos(α013 + ∠A3A1A), (2.27)
where
α013 = arccot
(sin(α213)− cos(α213) cot (α102)Co − (a13)0
(a12)0
cot (α103)Co
− cos(α213)− sin(α213) cot (α102)Co + (a13)0
(a12)0
)
,
(a10)0 =
sin (α013 + (α103)Co) (a13)0
sin (α103)Co
,
(αi0j)Co = arccos
(
(w′k)
2 − (w′i)
2 − (w′j)
2
2(w′i)(w
′
j)
)
,
for i, j, k = 1, 2, 3, i 6= j 6= k, and
ϕ0 =
√
1 + H2
r1
∠A1AA0(x00, (a10)g). (2.28)
492 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4
The Plasticity of Some Fittable Surfaces
R e m a r k 2. For given x00, α013 and (a10)0, the system of two nonlinear
equations (2.27) and (2.28) gives numerically w′1, w′2, taking into account that
w′3 = 1− w′1 − w′2.
E x a m p l e 1. Given are A1 = (cos 0, sin 0, 0), A2 = (cos π
3 , sin π
3 , 0.8),
A3 = (cos π
6 , sin π
6 , 2), on the circular cylinder x2 + y2 = 1.
The isometric mapping of the circular cylinder to R2 induced by (ϕ, z) yields
the points A′1 = (0, 0), A′2 = (π
3 , 0.8), A′3 = (π
6 , 2). Thus, the corresponding
Fermat–Torricelli point of 4A′1A
′
2A
′
3 F ′ = (0.8404027, 0.8536775) gives FC =
(0.667163, 0.744912, 0.8536775) for w1 = w2 = w3 = 1
3 .
Given A1, A2, A3, FC , we calculate the center of the fittable sphere x0, y0, z0, R
from the equations (2.7)–(2.14): x0 = −1.31848, y0 = −1.60442, z0 = 1.31278,
R = 3.11013.
From (2.17), (2.18), (2.19), (2.20), and (2.6), we derive that
(α102)S = 1.99478 rad,
(α203)S = 2.10237 rad,
(α103)S = 2.18604 rad,
which give
w′1 = 0.33281,
w′2 = 0.315291
and
w′3 = 0.3519
such that
w′1 + w′2 + w′3 = 1.
As a future work, we consider the following problem that may provide some
perspectives on the plasticity of geodesic triangles on some C2 complete surfaces
in R3 :
Problem 3. Suppose that F is the corresponding Fermat–Torricelli point of
a geodesic triangle 4A1A2A3 on a C2 complete surface M with positive weights
wi such that
w1 + w2 + w3 = 1.
Find a fittable Alexandrov surface M ′ of a bounded curvature which passes
from A1, A2 A3 and F such that F is the corresponding Fermat–Torricelli point
of 4A1A2A3 on M ′, with positive weights w′i, satisfying the equations
w′1 + w′2 + w′3 = 1
and w1 = w′1 or w1 = w′1 and w2 = w′2.
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4 493
A.N. Zachos
R e m a r k 3. If the points A1, A2, A3 are not fixed and belong to the surface
of a circular cylinder, then there is an isometric mapping which is deduced by
unrolling the circular cylinder by the line (generator of cylinder) which passes
from the weighted Fermat–Torricelli point A0 of 4 (4A1A2A3)C and the corre-
sponding weighted Fermat–Torricelli point of 4 (4A1A2A3)P on the Euclidean
plane coincides with A0, which yields w1 = w′1 and w2 = w′2.
We are interested in the derivation of non-isometric mappings of fittable sur-
faces which solve Problem 3, which could also lead to a new way of creating
two-dimensional fittable hyperbolic spaces (Plasticity of hyperbolic spaces).
Finally, we note that Problems 1 and 3 may provide an alternative characte-
rization of a Wald curvature ([7]) by placing geometric properties of the weighted
Fermat–Torricelli problem for geodesic triangles into Wald’s nonlinear quad.
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